Applying Quantum Representations of Mapping Class Groups (Wade Bloomquist, UCSB)

One foundational pillar of low dimensional topology is the connection between link invariants and 3-manifold invariants.  One generalization of this has been given by Reshetikhin and Turaev to a surgery theory for colored ribbon graphs.  Then to complete the analogy rather than 3-manifold invariants we now have a 2+1 dimensional topology quantum field theory (TQFT).  For this talk we will only be focusing on one corner of a TQFT, in particular the representations of mapping class groups which are afforded (called quantum representations).  We will first go through a brief construction of these representations, focusing on how colored ribbon graphs give rise to a basis.  Then we will dive into some applications of these representations both in recovering classical topology and in a proposal for a topological quantum computing protocol.  A strong effort will be made to keep things relatively self contained with as many pictures as possible.